Lecture 22 Highlights Phys 402

Lecture 22 Highlights Phys 402 Scattering experiments are one of the most important ways to gain an understanding of the microscopic world that is described by quantum mechanics. The idea is to take a known entity (for example an electron), give it a known energy and initial momentum (magnitude and direction), and send it on a collision course with an object whose structure and properties are not fully known. The known entity will interact with the particles making up the unknown substance through a (hopefully simple) interaction force. In the simplest experiments one then measures the energy and momentum of the known entity as it exits from the interaction region. It is assumed that the interactions take place only in a limited region around the target particles. This experiment is repeated many times for a given initial energy and momentum, and statistics of exiting energy and momentum are compiled. This exercise is repeated for other values of initial energy and momentum, resulting in a big data set. No wonder that the World Wide Web was invented by physicists trying to share this data with all of their colleagues around the world. Examples of scattering experiments include Rutherford scattering and angleresolved photoemission spectroscopy (ARPES), which is basically the photoelectric effect on steroids. Classical Scattering Theory The starting point for thinking about scattering is having a light particle incident from infinity on a heavy stationary target particle. Classically, the incident particle is travelling in a straight line as it approaches the potential created by the target particle. If it feels no interaction force, then it will travel by in an un-deviated straight line trajectory. The distance between its incident direction and the trajectory that sends it into a headon collision with the target particle is called the impact parameter and often denoted with the symbol bb. The extension of the head-on direction to infinity is defined as the z-axis. As the incident particle approaches the target it experiences a force that causes it to deviate away from its initial trajectory. After this interaction the light particle will be free once again of the target potential and move off in a straight line trajectory. We define this outgoing trajectory direction using spherical angular coordinates (θθ, φφ) from the above-defined z-axis. To describe the results of many such experiments with different impact parameters and outgoing directions, we establish a differential 1

relationship between a finite-size incident beam area and an outgoing beam of particles into a differential solid angle. The only quantity not controlled in a typical scattering experiment is the impact parameter bb of the projectile with respect to the target particle. The impact parameter is the distance of closest approach to the target particle, assuming no forces of interaction cause the projectile to change from its initial direction. Because we cannot control the impact parameter, we have to perform many experiments in which all possible values of bb are employed for the incident beam of projectiles. We then give a statistical description of the resulting scattering. With such a description, we can write the number of particles scattered NN ssssssssss in terms of the number of particles incident NN iiiiii as NN ssssssssss = NN iiiiii nn tttttttttttt σσ, where nn ttttrrgggggg is the density of target particles projected into the twodimensional plane (nn tttttttttttt ~1/mm 2 ) and σσ is defined as the scattering cross section of each particle. σσ is often measured in units of barns, which is 10 28 mm 2. We can generalize the concept of cross section to any process, including capture (σσ cccccccccccccc ), ionization (σσ iiiiiiiiiiiiiiiiiiii ), fission (σσ ffffffffffffff ), etc. This is done by using the definition NN ssssssssss,xx = NN iiiiii nn tttttttttttt σσ xx for process xx. Experiments start with a beam of projectile particles of identical structure and equal initial momenta and energy. The projectiles enter the target with all possible values of impact parameter. One then measures how many particles come out with angle of exit θθ, φφ and also the energy and momentum of the exiting particle. Our job is to identify the force of interaction between the projectile and target particles from the number of particles scattered through angle θθ, φφ, for all possible angles. We write the angleresolved scattering cross section as dddd dddd NN ssssssssss (iiiiiiii aaaaaaaaaaaa θθ, φφ) = NN iiiiii nn tttttttttttt (θθ, φφ), where (θθ, φφ) is called the differential scattering cross section (DSCS). Note that the element of differential solid angle is = sin θθ dddddddd. We expect that if this quantity is integrated over all possible exiting angles, we should recover the total scattering cross section for this process: σσ = dddd (θθ, φφ). We shall assume that all scattering potentials are spherically symmetric, hence there will be no dependence on the φφ coordinate. To find dddd (θθ, φφ) we compare the area covered by the incident particles at impact parameters between bb and bb + dddd in an angle dddd (i.e. dddd = bb dddd dddd) to the solid angle subtended by the exiting beam of particles (i.e. = sin θθ dddd dddd) to arrive at dddd bb sin θθ dddd dddd =. To find the DSCS, we need to calculate the trajectory of a projectile particle for every possible impact parameter. We then did the example of a point particle elastically scattering from a fixed hard sphere of radius RR and found that bb = RR cos θθ dddd, = RR2, 2 4 which is independent of angle! The total scattering cross section is just σσ = ππrr 2, which is the cross-sectional area presented by the sphere. Another famous case of classical scattering theory is Rutherford scattering associated with a Coulomb interaction. This type of scattering experiment was used to deduce that most of the mass of the atom was concentrated in a small volume known as the nucleus. The DSCS was deduced to be : DD(θθ) = dddd = qqqq/4ππεε 0 4EE sin 2 (θθ/2) 2, where EE is the energy of the 2

incident particle of charge qq approaching a target of charge QQ and scattering through 1 angle θθ. The DSCS has a distinctive angular dependence, which was clearly sin 4 (θθ/2) observed by Geiger and Marsden using αα-particles scattering from thin Au foils. They also showed that the scattering rate scaled with nn tttttttttttt (by varying the thickness of the foil), scaled as 1/E 2 (by varying the energy of the incident alpha particles), scaled as 1 (by measuring the number of particles scattered vs. outgoing angle), and scaled sin 4 (θθ/2) as Z 2, where Q=+Ze is the nuclear charge. Note that because dddd ~qq2 QQ 2, the scattered particle distribution is insensitive to whether the Coulomb interaction is attractive or repulsive. Also, the agreement for the angular dependence of dddd with data suggests that the Coulomb force has the simple 1/r2 dependence even down to nuclear length scales. Finally, the total scattering cross section calculated from this dddd diverges. This is because the bare Coulomb force is infinitely long ranged. In reality, the Coulomb force of the nucleus is screened out by the electron cloud of the atom, on the length scale of one nm, or less. When this screening is taken into account the total scattering cross section becomes finite, as observed. These calculations assume that the alpha particle only undergoes one scattering event in the material (the Born scattering approximation). In addition, because of the electron screening, when an alpha particle is near one nucleus, it is insensitive to all the other nuclei because they are cloaked by their neutralizing electron clouds. Quantum Scattering Theory In quantum mechanics one does not describe the particles in terms of trajectories, but as waves. We replace the incident particle trajectory with a plane wave moving in the z-direction ψψ iiiiiiiiiiiiiiii = AA ee iiiiii, where the energy of the particle is EE = ħ 2 kk 2 /(2mm). This wave interacts with the scattering center and sends out a collection of outgoing spherical waves centered on the scatterer, as shown in the picture above. We expect solutions at large distances from the scattering center of the form, 3

ψψ(rr, θθ) = AA ee iiiiii + ff(θθ) eeiiiiii, (1) rr where the first term in the bracket is the incoming plane wave and the second term is an outgoing spherical wave with a direction-dependent coefficient ff(θθ). Since the scattering potential is assumed to be spherically symmetric we expect no dependence on the azimuthal spherical coordinate, φφ. Instead of calculating a trajectory and an impact parameter for this wave (which makes no sense) we instead deal with probability flow through the scattering region. Equating the probability of the incident particle being in a differential volume of size dddd vv dddd (where vv is the speed) to the outgoing scattered wave being in a differential volume rr 2 vv dddd described by the wavefunction above, yields the DSCS DD(θθ) = dddd = ff(θθ) 2. One can proceed using Partial Wave Analysis. This process decomposes the scattering by breaking it into a series of scattering events of progressively higher angular momentum scattering states. This is essentially analogous to considering classical particles with larger and larger values of the impact parameter. We wish to solve the full Schrodinger equation for a spherically symmetric potential VV(rr). From previous studies of the 3D Schrodinger equation in chapter 4 we know the successful ansatz for this case is ψψ(rr, θθ, φφ) = RR(rr) YY l,mm (θθ, φφ), where YY l,mm (θθ, φφ) are the spherical harmonics. This separates the problem into angular equations (already solved by the spherical harmonics) and a radial equation. Defining uu(rr) rrrr(rr), we find that the radial equation reduces to ħ2 dd 2 uu 2mm ddrr l(l+1)ħ2 + VV(rr) + uu = EEEE (2) 2 2mmrr2 We divide the problem into 3 regions. First is the asymptotic region where both VV(rr) 0 and the centrifugal term l(l+1)ħ2 can be ignored. In this Radiation zone one 2mmrr2 has kkkk 1. The radial Schrodinger equation becomes quite simple: ħ2 dd 2 uu = EEEE, or dd2 uu = ddrr 2 ddrr 2 kk2 uu, with solutions uu(rr) ~ ee ±iiiiii. This gives an outgoing 2mm wave of the form RR(rr) = uu eeiiiiii = AA, which was the form posited above in Eq. (1). rr rr The next domain is the Intermediate region where we can assume that VV(rr) 0 but the centrifugal term cannot be neglected. This assumes that the scattering potential is localized, which means effectively that it falls off faster than 1 rr2. In this region the radial Schrodinger equation becomes ħ2 dd 2 uu uu = EEEE, or dd2 uu uu = kk 2 uu. The solutions are of the form 2mm ddrr 2 + l(l+1)ħ2 2mmrr 2 ddrr 2 l(l+1) rr 2 of spherical Bessel functions: uu(rr) = AA rrjj l (kkkk) + BB rrnn l (kkkk), where the jj l (xx) remain finite as xx 0, while the nn l (xx) do not. Roughly speaking jj l (kkkk) is analogous to sin xx, while nn l (kkkk) is analogous to cos xx. As with trigonometric functions, it is sometimes useful to consider their complex sum, which are called the spherical Hankel functions of the first and second kind, defined as h l (1) (kkkk) = jj l (kkkk) + iinn l (kkkk) and h l (2) (kkkk) = jj l (kkkk) iinn l (kkkk). The spherical Hankel function of the first kind has the useful property that it turns into an outgoing spherical wave for large argument: h l (1) (kkkk 1) eeiiiiii rr. The solution for the radial wavefunction now becomes RR(rr) = h l (1) (kkkk). To allow for all 4

possible impact parameters we have to allow for every possible angular momentum quantum number in the solution to the angular equation (note that classically the angular momentum of the incoming particle is LL = mmmmmm, which means that a sum over l is roughly analogous to a sum over impact parameters). This means the solution is in the form of an infinite sum: ψψ(rr, θθ) = AA ee iiiiii + l,mm CC l,mm h (1) l (kkkk) YY l,mm (θθ, φφ), where the CC l,mm are unknown expansion coefficients. However, because we shall assume that the scattering potential is azimuthally symmetric, only the mm = 0 terms are relevant to the expansion. Since YY l,mm (θθ, φφ)~ee iiiiii, it effectively reduces to just the Legendre polynomials as a function of θθ: YY l,mm=0 (θθ, φφ) = 2l+1 4ππ PP l(cos θθ). Re-writing the expansion coefficients as CC l,0 = ii l kk 4ππ(2l + 1) aa l (to make it compatible with a future expression for the incoming wave), where we have now defined the unknown complex partial wave amplitudes aa l. The solution to the scattering problem can now be written as ψψ(rr, θθ) = AA ee iiiiii + kk ii l (2l + 1) aa l h (1) l l (kkkk) PP l (cos θθ). By looking in the limit kkkk 1 and using the asymptotic form for the spherical Hankel function, we find the solution reduces to the form of Eq (1) with the scattering amplitude ff(θθ) = l=0 (2l + 1) aa l PP l (cos θθ), written in terms of the partial wave amplitudes. This allows us to write the total scattering cross section as σσ = DD(θθ) = ff(θθ) 2 = 4ππ l=0 (2l + 1) aa l 2. The total cross section is thus simply related to the weighted sum of the absolute squares of the partial wave amplitudes. To find these amplitudes we need to solve the full Schrodinger equation (Eq. (2)) including the scattering potential VV(rr) in the Interior region and match the boundary conditions with the intermediate and radiation zone solutions. A technical step is taken to re-express the ee iiiiii incoming wave with Rayleigh s formula as a sum over all angular momenta, so that the full solution becomes: ψψ(rr, θθ) = AA ii l (2l + 1) jj l (kkkk) + ii kk aa l h (1) l l (kkkk) PP l (cos θθ). As an example we considered the quantum version of hard-sphere scattering. ffffff rr aa This is a potential described by VV(rr) =. The solution to the full 0 ffffff rr > aa Schrodinger equation is pretty straightforward in this case. We simply require hard sphere boundary conditions, namely ψψ(aa, θθ) = 0. Due to the independence of each jj term in the sum on l, each term must separately be zero, yielding aa l = ii l (kkkk) (1). The kk h l (kkkk) total scattering cross section can be written as σσ = 4ππ (2l + 1) aa l 2 4ππ l=0 = kk 2 2 (1) h l (kkkk) (2l + 1) jj l (kkkk) l=0. This expression is not particularly informative. However, consider the small sphere limit kkkk 1, which means that the sphere is much smaller than the wavelength of the incident particle, or alternatively the incident particle has low energy. By examining the Bessel functions in the small argument limit, one arrives at the further un-helpful result: σσ = 4ππ 1 kk 2 l=0 2l l! 2l+1 (2l)! 4 (kkkk) 4l+2. But in the kkkk 1 limit we can take just the first 5

term in the sum since kkkk appears to such a high power, leading to σσ 4ππaa 2, which corresponds not to the cross-sectional area presented by the sphere, but rather its entire surface area! The quantum waves somehow feel their way around the entire sphere during the scattering process. 6